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VOL. 27,
NO. 5, MAY 2015
1167
A General Geographical Probabilistic Factor
Model for Point of Interest Recommendation
Bin Liu, Hui Xiong, Senior Member, IEEE, Spiros Papadimitriou, Yanjie Fu, and Zijun Yao
Abstract—The problem of point of interest (POI) recommendation is to provide personalized recommendations of places, such as
restaurants and movie theaters. The increasing prevalence of mobile devices and of location based social networks (LBSNs) poses
significant new opportunities as well as challenges, which we address. The decision process for a user to choose a POI is complex
and can be influenced by numerous factors, such as personal preferences, geographical considerations, and user mobility behaviors.
This is further complicated by the connection LBSNs and mobile devices. While there are some studies on POI recommendations, they
lack an integrated analysis of the joint effect of multiple factors. Meanwhile, although latent factor models have been proved effective
and are thus widely used for recommendations, adopting them to POI recommendations requires delicate consideration of the unique
characteristics of LBSNs. To this end, in this paper, we propose a general geographical probabilistic factor model (Geo-PFM)
framework which strategically takes various factors into consideration. Specifically, this framework allows to capture the geographical
influences on a user’s check-in behavior. Also, user mobility behaviors can be effectively leveraged in the recommendation model.
Moreover, based our Geo-PFM framework, we further develop a Poisson Geo-PFM which provides a more rigorous probabilistic
generative process for the entire model and is effective in modeling the skewed user check-in count data as implicit feedback for better
POI recommendations. Finally, extensive experimental results on three real-world LBSN datasets (which differ in terms of user
mobility, POI geographical distribution, implicit response data skewness, and user-POI observation sparsity), show that the proposed
recommendation methods outperform state-of-the-art latent factor models by a significant margin.
Index Terms—Recommender systems, point of interest (POI), probabilistic factor model, location-based social networks
Ç
1
INTRODUCTION
R
ECENT years have witnessed the increased development
and popularity of location-based social network
(LBSN) services, such as Foursquare, Gowalla, and
Facebook Places. LBSNs allow users to share their check-ins
and opinions on places they have visited, ultimately helping
each other find better services. Data collected through LBSN
activity can enable better recommendations of places, or
Points of Interest (POIs) such as restaurants and malls. This
can drastically improve the quality of location-based
services in LBSNs, simultaneously benefiting not only
LBSN users but also POI owners. On one hand, mobile users
can identify favorite POIs and improve their user
experience via good POI recommendations. On the other
hand, POI owners can leverage POI recommendations for
better targeted acquisition of customers. In this paper we
address exactly the problem of POI recommendation. We
first identify the key challenges specific to geographical
settings. Then, we propose a general framework to address
these, as well as two instantiations of this framework.
Challenges. While latent factor models, such as matrix factorization [19], probabilistic matrix factorization (PMF) [27],
[28], and many other variants [1], [3], [17], [18], [22], [36],
The authors are with the Department of Management Science and
Information Systems, Rutgers Business School, Rutgers University,
Piscataway, NJ 08854.
E-mail: {binben.liu, hxiong, s.papadim, yanjie.fu, zijun.yao}@rutgers.edu.
Manuscript received 4 May 2014; revised 22 Sept. 2014; accepted 24 Sept.
2014. Date of publication 8 Oct. 2014; date of current version 27 Mar. 2015.
Recommended for acceptance by J. Wang.
For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference the Digital Object Identifier below.
Digital Object Identifier no. 10.1109/TKDE.2014.2362525
have been proved effective and are widely used in diverse
recommendation settings, adapting them to POI recommendations requires delicate consideration of unique characteristics of LBSNs. Indeed, there are several characteristics of
LBSNs which distinguish POI recommendation from traditional recommendation tasks (such as movie or music
recommendations). More specifically:
Geographical influence. Due to geographical constraints and the cost of traveling large distances, the
probability of a user visiting a POI is inversely proportional to the geographic distance between them.
Tobler’s first law of geography. The law of geography
states that “Everything is related to everything else,
but near things are more related than distant things”
[32]. In other words, geographically proximate POIs
are more likely to share similar characteristics.
User mobility. Users may check into POIs at different
regions; e.g., an LBSN user may travel to different
cities. Varying user mobility imposes huge challenges on POI recommendations, especially when a
user arrives at a new city or region.
Implicit user feedback. In the study of POI recommendations, explicit user ratings are usually not
available. The recommender system has to infer user
preferences from implicit user feedback (e.g., checkin frequency).
The first three mutually related challenges due to
geography imply interrelationships among items. However,
traditional recommender systems usually ignore these,
assuming that the items are independent and identically
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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING,
distributed. In fact, the decision process of a user choosing a POI is complex and can be influenced by many factors. First, geographical distance plays an important role.
According to the Tobler’s first law of geography and the
law of demand, a user’s propensity for a POI is inversely
proportional to the distance between them. This is similar
to the observation that the probability of purchasing an
item is inversely proportional to its cost. Second, utility
matters. In economics, utility is an index of preferences
over sets of items and services when a user makes purchasing decisions. In other words, a user may still prefer
a remote POI to a nearby one, if higher satisfaction (utility) outweighs the overhead of travel. Finally, LBSN
users have varying mobility behaviors, which further
impose challenges on modeling check-in decisions.
An additional fourth challenge is that user check-in
counts follow a distribution with power-law form. This is
different from ratings in traditional recommender systems,
in which explicit ratings are available to reflect users’ item
preferences. In other words, in LBSNs a user can visit a POI
only once and another POI hundreds of times. Since we do
not have explicit user ratings for POIs, we can only make
use of implicit user behavior data in the check-in records for
POI recommendations.
POI recommendation framework. All the above challenges
demand a reconsideration of the recommendation model, to
achieve effective POI recommendation in LBSNs. While
there are some studies on POI recommendations, they lack
an integrated analysis of the joint effect of the above factors,
such as user preferences, geographical influences and user
mobility behaviors.
To address these challenges, we propose a framework
for geographical probabilistic factor modeling (Geo-PFM)
which can strategically take various factors into consideration. This framework can capture the geographical influences on a user’s check-in behaviors, can effectively
model the user mobility patterns, and can deal with the
skewed distribution of check-in count data. Specifically,
we introduce a latent region variable and use a multinomial distribution over latent regions to model user mobility behaviors over different activity regions. These latent
regions reflect the activity areas for all the users through
collective actions. A Gaussian distribution is used to represent a POI over a sampled region. This can reflect the
first law of geography; that is, similar POIs are more
related than distant POIs. Moreover, geographical influence can be effectively modeled in the latent region.
Finally, implicit user feedback in the form check-in counts
is taken into account.
In our earlier work [23], we introduced Geo-PFM by specifically instantiating a geographical Bayesian non-negative
matrix factorization(Geo-BNMF), to model user preferences. As a result, this model is capable of taking personal
preferences, geographical influence, and user mobility into
consideration, and can effectively handle the skewed distribution of POI count data.
In this paper, we study the Geo-PFM framework in
more detail and we further develop a Poisson Geo-PFM,
which is also able to capture the geographical influences
on a user’s check-in behavior and effectively model the
VOL. 27,
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MAY 2015
user mobility patterns. In addition, the Poisson Geo-PFM
provides much more flexibility and interpretability than
Geo-PFM based on non-negative matrix factorization
[23]. First, the Poisson Geo-PFM provides a rigorous
probabilistic generative process for the model, while the
NMF-based Geo-PFM uses an approximation solution.
Second, the nature of Poisson distribution is more suitable and effective for modeling the skewed user check-in
count data, which provide implicit feedback, for better
POI recommendations.
Finally, we provide extensive experimental results on
three real-world LBSNs data, which differ in terms of
user mobilities, POI geographical distributions, implicit
response data skewness and user-POI observation sparsity. The experimental results show that the proposed
POI recommendation method consistently outperforms
state-of-the-art probabilistic latent factor models with a
significant margin in terms of Top-N recommendation.
Moreover, the proposed Poisson Geo-PFM outperforms
Geo-BNMF [23] even further.
2
BACKGROUND
Latent factors models aim to characterize user-item interactions assuming that each user and each item can be
expressed as a user and item latent vector u i and v j respectively. Consequently, the response (rating,
like, or implicit
frequency) is modeled as pðyij j i; jÞ ¼ p yij j u >
i v j ; Q : In this
section we summarize two types of latent factor models:
probabilistic matrix factorization methods which are widely
used for recommendations when explicit user feedback
(e.g., item ratings) is available, and the Poisson factor model
(PoiFM) which is more effective when user feedback is
implicitly provided via heavily skewed frequency counts
(as in the case of POI recommendation).
2.1 Probabilistic Matrix Factorization
Matrix factorization models [19] have been generalized into
probabilistic matrix factorization [28], which is a Bayesian
version. In PMF the response yij of user ui for item vj is
assumed to follow a Gaussian distribution yij N ðyij j
2
u>
i v j ; s Þ. When response yij is not normalized to a standard
rating score, one solution is to scale the discrete response to
a value between ð0; 1 by using fðxÞ ¼ ðx 1Þ=ðxmax 1Þ,
where xmax is the maximum response value for each user
[28]. Furthermore, a zero-mean Gaussian prior is placed
toon the user and item latent spaces
M
Y
N u i j 0; s 2u I ;
P U j s 2u ¼
i¼1
P V j s 2v ¼
N
Y
N v j j 0; s 2v I :
j¼1
Then the latent factors u and v can be inferred by maximize thing likelihood over the observed ratings
P ðY j U; V; s 2 Þ ¼
M Y
N Y
2 Iij
N yij j u >
;
i vj ; s
(1)
i¼1 j¼1
where Iij is the indicator function. Maximizing the logposterior over user and item latent factors with
LIU ET AL.: A GENERAL GEOGRAPHICAL PROBABILISTIC FACTOR MODEL FOR POINT OF INTEREST RECOMMENDATION
hyperparameters is equivalent to minimizing the sum-ofsquared-errors objective function:
M X
N
2
1X
Iij yij u>
L¼
i vj
2 i¼1 j¼1
(2)
M
N
U X
V X
2
2
þ
jju
ui jjF þ
jjvvj jjF ;
2 i¼1
2 j¼1
where U ¼ s 2 =s 2u , V ¼ s 2 =s 2v , and jj jj2F is the Frobenius
norm. Gradient descent can be applied to infer the latent
factors with partial derivatives u i and v j respectively,
N
X
@L
¼
Iij yij u>
i v j v j þ U u i
@u
ui
j¼1
M
X
@L
¼
Iij yij u>
i v j u i þ V v j :
@vvj
i¼1
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TABLE 1
Mathematical Notations
Symbol
R
h
m
S
U
V
yij
lj
(3)
Size
Description
1 jRj
M jRj
R2
R22
MK
N K
R
R2
latent region set, r is a region in R
user level region distribution
location mean of a latent region
covariance matrix of a latent region
user latent factor
item latent factor
response of user i for item j
location of item j
LðU; V ; j D; aU ; bU ; aV ; bV Þ
¼
M X
K X
ðaU 1Þ ln uik uik =bU
i¼1 k¼1
2.2 Poisson Factor Model
The Poisson distribution is a more appropriate choice for
response variables yij that represent frequency counts. The
Poisson probabilistic factor model (Poi-PFM) [6], [12], [26]
factorizes the user-item count matrix Y as Y PoissonðUV Þ:
More specifically, for each user-item response yij , we
assume a Poisson distribution over the mean fij : yij Poissonðfij Þ. The mean matrix F is factorized into two matrices U MK and V NK . Each element uik 2 U encodes the
preference of user i for “topic” k, and each element vik 2 V
reflects the topical affinity of item j to topic k. Further, uik
and vik can be assigned empirical priors following Gamma
distributions. We then have the following generative
process.
1. Generate user latent factor uik GammaðaU ; bU Þ:
2. Generate item latent factor vjk GammaðaV ; bV Þ:
u>
3. Generate yij Poissonðu
i v j Þ:
Given user latent factor u i and item latent factor v j , the
probability of response yij is
yij
P ðyij j u i ; v j Þ ¼ u>
exp u >
i vj
i v j =yij !:
We can apply maximum a posteriori (MAP) estimation
over the observed data and priors to infer the latent vectors.
Specifically,
P ðU; V j Y; aU ; bU ; aV ; bV Þ
/ pðY j U; V ÞP ðU j aU ; bU ÞpðV j aV ; bV Þ;
where
pðY j U; V Þ ¼
M Y
N Y
yij
u>
exp u >
i vj
i v j =yij !
i¼1 j¼1
P ðU j aU ; bU Þ ¼
a 1
M Y
K
Y
u U expðuik =bU Þ
ik
i¼1 k¼1
pðV j aV ; bV Þ ¼
a
bUU GðaU Þ
a 1
N Y
K
Y
u V expðvjk =bV Þ
ik
j¼1 k¼1
a
bVV GðaV Þ
:
The log of the posterior distribution over the user and item
latent factors is given by
þ
N X
K X
ðaV 1Þ ln vjk vjk =bV
(4)
j¼1 k¼1
þ
M X
N
X
ðyij ln fij fij Þ þ const:
i¼1 j¼1
Taking derivatives on L with respect to uik and ujk , we
have
N X
@L
aU 1
1
yij
¼
þ
1 vjk
@uik
uik
bU j¼1 fij
M X
@L
aV 1
1
yij
¼
þ
1 uik :
@vjk
vjk
bV
fij
i¼1
(5)
Again, gradient ascent method can be applied to infer the
latent factors.
3
GEOGRAPHICAL PROBABILISTIC FACTOR MODEL
FOR POI RECOMMENDATION
In this section, we first formulate the problem of POI recommendation and then introduce a general geographical probabilistic factor analysis framework for this problem,
addressing the challenges described previously.
3.1 Problem Definition
The problem of personalized POI recommendation is to recommend POIs to a user given user POI check-in records
and other available side information. Let U ¼ fu1 ;
u2 ; . . . ; uM g be a set of LBSN users, where each user has a
location li . The user location li is usually unknown due to
user mobility. Let V ¼ fv1 ; v2 ; . . . ; vN g be a set of POIs,
where each POI has a location lj ¼ ½lonj ; latj > represented
by longitude and latitude. Throughout the paper we use
indices i for users and indices j for POIs, unless stated otherwise. The number of times user ui visited POI vj is represented by the response variable yij . The check-in records for a
particular user are sparse (most yij values are zero), with
non-zeros following a power law. In the paper we use the
terms “POI” and “item” interchangeably. Key notations are
listed in Table 1.
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Fig. 1. An example of a typical user check-in pattern: (a) all the POIs; (b) the user’s check-ins over different regions: San Francisco, Los Angeles, San
Diego, Las Vegas, Houston, and New York City; (c) the user’s check-ins in San Francisco area.
3.2 The General Idea
We aim to capture how different factors such as user preference, geographical influence and user mobility affect user
POI check-in decisions. The key idea is that overall user
preferences are the result of the interplay between geographical preferences and interest preferences. Our models
aim to effectively capture that interplay.
Geographical preferences. To learn geographical user
preferences, we need a model to encode the spatial influence and user mobility into the user check-in decision
process. As shown in Fig. 1, LBSN users are most likely
to check into a number of POIs and these POIs are usually
limited to certain geographical regions. This observation
has two implications: first, a user’s mobility always happens across a limited number regions but these regions
could be different among different users; second, user
check-in activities happen in a given region and the activity patterns could be different given different regions.
Based on this observation, we propose to introduce a set
of jRj latent regions R which are inferred based on the
collective actions of all users, reflecting activity areas for
the entire population.
Although the overall distribution of POIs is irregular, we
can however assume a Gaussian geographical distribution
of POIs within each region r 2 R. The location lj for POI j is
characterized by lj N ðmr ; Sr Þ, where mr and Sr are the
mean vector and covariance matrix of the region, respectively [14], [35]. Latent regions also reflect Tobler’s first law
of geography, which states that POIs with similar
characteristics are likely to be clustered into the same geographical area. Once a region is fixed, geographical influence can be effectively modeled and applied to overall user
preference profiling.
We finally model individual user mobility over the collectively inferred latent regions R by applying a multinomial distribution, r pðr j hi Þ, where hi is a user-dependent
distribution over latent regions for user i.
Interest preferences. Interest preferences are modeled
using a latent factor model, generating a user item preference aði; jÞ based on user latent factor variable u i and and
item latent factor variable v j .
Overall user preferences. Finally, to model a user’s propensity for a POI, we assume the following factors that will
affect the overall user check-in decision process: (1) the personal preference aði; jÞ of each user i with respect to POI j;
and (2) geographical influence in terms of travel distance,
namely, the distance dði; jÞ between the user and the POI as
a geographical cost. As a result, the probability of observing
a user-POI pair ði; jÞ is directly proportional to the user
interest, and monotonically decreases with the distance
between them,
pði; jÞ / F aði; jÞ;
d0
d0 þ dði; jÞ
t ;
where FðÞ is a function that combines user interest preference and geographical influence. We model the distance factor in the decision making process using a
d0
parametric term ½d þdði;jÞ
t with a power-law form. This
0
motivated by the observation that the probability of user
i choosing POI j decays exponentially with respect to the
distance between them.
3.3 Geographical Probabilistic Factor
Model Framework
Based on above discussion, we proposed a geographical
probabilistic factor model to capture user mobility, and geographical influence in user profiling for POI recommendation. The complete graphical model is shown in Fig. 2.
The corresponding generative process to draw pairs ði; jÞ
representing user i choosing POI j can then be expressed as
follows. First, a user ui samples a region ri from all jRj
regions following a multinomial distribution ri Multinomialðhi Þ, on which a conjugate Dirichlet prior Dirðgg Þ
can be further imposed. Here hi is a user-dependent parameter, capturing user i’s mobility pattern over the latent
Fig. 2. A graphical representation of the proposed geographical probabilistic factor model, where the red plate represents users, the blue plate
represents POIs, and the purple plate represents latent regions. The
model priors have been excluded for simplicity.
LIU ET AL.: A GENERAL GEOGRAPHICAL PROBABILISTIC FACTOR MODEL FOR POINT OF INTEREST RECOMMENDATION
regions. A POI is drawn from the sampled region lj N ðmri ; Sri Þ. The interest preference aði; jÞ of user i for POI j
can be represented by combining latent factors u i and v j ,
specifically, aði; jÞ ¼ u >
i v j . Finally, the user-POI response yij
(check-in frequency count) is assumed to follow certain
distribution yij P ðfij Þ where fij depends on user preferences and the distance between the user and the POI.
Summarizing:
1.
2.
3.
Draw a geographical preference
a. Draw region ri Multinomialðhi Þ:
b. Draw a POI j with location lj N ðmri ; Sri Þ.
Draw an interest preference
a. Draw user latent factor u i P ðu
ui ; Cu i Þ:
b. Draw item latent factor v j P ðvvj ; Cv j Þ:
c. Draw user-item preference aði; jÞ ¼ u >
i vj :
For each user-POI pair ði; jÞ draw the response
yij P ðfij Þ, where
fij ¼ F u >
v
;
j
i
d0
d0 þ dði; jÞ
influence can be effectively captured by the proposed
Geo-PFM model.
3.4.2 Modeling Count Response
In most existing latent
represented by PMF
factor models,
v
;
Q
is
assumed
to follow a
[28], the response P yij j u >
i j
2
u>
v
;
s
Þ.
However,
Gaussian distribution, namely, yij N ðu
i j
a Gaussian distribution is not suitable when the response
variable is implicit count data, which are heavily skewed.
Therefore, it is not suitable for the POI recommendation
problem, since check-in counts follow a power-law like
distribution.
We need to ensure our model is suitable for count
responses. By combining geographical influence with latent
factors, we model user-POI response as a geographical
probabilistic factor model:
h
it d0
:
yij P ðyij j fij ; QÞ; fij ¼ F u >
i v j ; d0 þdði;jÞ
t :
Note that the proposed model is general and can be
extended with different factor models, since we limit neither the user and item latent factor distribution, nor the
user-item response distribution. FðÞ is a function of persond0
t
alized preferences u>
i v j and of distance cost ½d0 þdði;jÞ . Useritem response yij P ðfij Þ can be: (i) Gaussian when explicit
ratings are available, (ii) Bernoulli for binary response such
as liking, or (iii) Poisson when count or frequency data is to
be modeled.
3.4 Model Components
This section describes the model components of Geo-PFM
in detail.
3.4.1 User Mobility and Geographical Influence
As discussed earlier, user mobility and geographical influence are among the most predominant factors that
distinguish POI recommendation from traditional recommendation (e.g., for movies), and these two factors can
interact with each other. Geographical influence has been
exploited for POI recommendation due to the fact that geographical proximity could significantly affect a user’s
check-in decision [34]. However, check-in behavior can
change as the user travels from one region to another, and
little has been done to consider user mobility for POI recommendation. Capturing user mobility is important to understand user preferences in different regions, and it becomes
even more important when a user travels to a new place.
To this end, as described earlier, we introduce a set of jRj
latent regions R, and model user mobility using multinomial
distribution [14] r Multinomialðhi Þ, where hi is a userdependent distribution over latent regions for user i. The
explicit location ‘ðÞ of a user is not observed. We use the
region r with center mr to represent the user activity area and
model the geographical influence as a parametric and
d0
power-law like term ½d0 þdði;jÞ
t , with dði; jÞ ¼ jjmr lj jj2 ,
where mr approximates the current user activity area
center. As a result, both user mobility and geographical
1171
In the above, FðÞ is a suitably chosen function that captures
the joint effect of personal interest preferences u>
i v j and disd0
t . Also, the response function P ðÞ suittance cost ½d0 þdði;jÞ
ably chosen to model count data. Potential response
function distributions include Poisson (see Section 4).
4
MODEL SPECIFICATION
This section introduces detailed model specifications of the
Geo-PFM model. In particular, we introduce a Poisson
Geo-PFM model, which takes into account the characteristics of count response values.
4.1 Poisson Geo-PFM Model
As we use count response to infer user preferences, we
expect the latent vectors are constrained to be non-negative.
In our earlier work [23], we applied a rectified normal
Bayesian non-negative matrix factorization model to capture the count response feature. Specifically, we assumed a
rectified normal distribution on Y P ðU
U V Þ with variance
s 2 I and non-negativity constraints,
Y N R ðY j U V ; s 2 IÞ;
subject to U 0; V 0:
(6)
We further placed an exponential distribution on U and V ,
and an inverse gamma distribution on s 2 with shape a and
scale b.
However, a Poisson factor model is a better alternative.
First, the Poisson distribution is a more appropriate choice
for modeling skewed count data. Fig. 3 shows a typical distribution of check-in count distribution, for a randomly
selected user in the Foursquare dataset. A Poisson distribution approximates this distribution well, and can also provide a response that is non-negative. More importantly, a
Poisson factor guarantees a rigorous probabilistic generative process for the model, while the rectified normal BNMF
provides a probabilistic approximation. Therefore we propose a Poisson Geo-PFM model which incorporates both
user interest preference and geographical influence. More
specifically, for each user-item frequency yij we assume a
Poisson distribution over mean fij : yij Poissonðfij Þ with
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4.2.1 E-step
In the E-step, we iteratively draw latent region assignments
for all POIs. For each POI, a latent region r is first drawn
from the following distribution:
(7)
r P r j yj ; lj ; RðtÞ ; CðtÞ P ðr j h ðtÞ Þ;
where
P yj ; lj j r; CðtÞ ¼ P ðlj j r; CðtÞ Þ P ðyj j r; CðtÞ Þ
ðtÞ
P ðlj j r; CðtÞ Þ ¼ N ðlj j mðtÞ
r ; Sr Þ
Fig. 3. The check-in counts distribution of a randomly selected user and
a Poisson approximation of this distribution (Foursquare dataset).
d0
t
fij ¼ u >
i v j ½d0 þdði;jÞ . Furthermore, uik and vik are given
Gamma distributions as empirical priors [6], [26], uik GammaðaU ; bU Þ and vjk GammaðaV ; bV Þ. Then, the generative process we introduced earlier to model user-item preference becomes specifically:
1.
2.
3.
4.
5.
Draw a region r Multinomialðhi Þ:
Draw a POI j with location lj N ðmr ; Sr Þ.
Draw user latent factor uik GammaðaU ; bU Þ:
Draw item latent factor vjk GammaðaV ; bV Þ:
Draw yij Poissonðfij Þ with
t
d0
fij ¼ u >
v
:
j
i
d0 þ dði; jÞ
P ðyj j r; CðtÞ Þ ¼
P ðr j h ðtÞ Þ updates region assignment in terms of user mobility, P ðlj j r; CðtÞ Þ is the location PDF function for multivariate
normal distribution with mean vector and variance matrix
obtained in last iteration, and P ðyj j r; CðtÞ Þ updates region
assignment through collective actions.
4.2.2 M-step
In the M-step, we maximize the log likelihood of the model
with respect to model parameters by fixing all regions
obtained in the E-step. Since we sample the regions in the
E-step, we can update mr ; S r ; h directly from the samples,
D
1 X
Iðrj ¼ rÞlj
#ðj; rÞ j¼1
D
X
1
Sr ¼
ððlj mr Þðlj mr Þ> Þ
#ðj; rÞ 1 j¼1
Y P yij ; lj ; C j V P ðU j a; bÞP ðV j a; bÞP ðh j gÞ
D
(N
M
Yi
Y
1
12
T 1
jS
Sr j exp ðlj m r Þ Sr ðlj m r Þ
/
2
i¼1 j¼1
M Y
K
Y
fij yij expðfij Þ
uik
ciR
ci1
a1
hi1 hiR uik exp
yij !
b
i¼1 k¼1
N Y
K
M Y
R
Y
Y
ujk
g 1
va1
hiri :
jk exp
b
j¼1 k¼1
i¼1 r¼1
To estimate the parameters C, we use a mixing Expectation Maximization (EM) and sampling algorithm to learn all
the parameters [2], [14]. We regions r as a latent variable
and introduce the hidden variable P ðr j lj ; CÞ [14], [35],
which is the probability of lj 2 r, given POI location lj and
C. The algorithm iteratively updates the parameters by
mutual enhancement between Geo-clustering and
Geo-PFM. The Geo-clustering updates the latent regions
based on both location and check-in behaviors; and
Geo-PFM learns the graphical preference factors.
(8)
where #ðj; rÞ is the number of POIs assigned to region r.
Through imposing a conjugate Dirichlet prior Dirðgg Þ, we
update h ðtþ1Þ by
ðtþ1Þ
hir
D
/
Poissionðfij j U ðtÞ ; V ðtÞ Þ:
i¼1
mr ¼
4.2 Parameter Estimation
Let C ¼ fU
U ; V ; h ; m ; Sg denote all parameters, and let
V ¼ faU ; bU ; aV ; bV ; gg be the hyperparamters. We are given
I
the observed data collection D ¼ yij ; lj ij where yij is the
user check-in count and lj is the location of vj ; and Iij is the
indicator function with Iij ¼ 1 when user ui visited POI vj ,
and Iij ¼ 0 otherwise. Then we aim to maximize the posterior probability given the observed data:
Y P ðC; D; VÞ /
P yij ; lj j C; V P ðC j VÞ
M
Y
ðtþ1Þ
¼
Cir
ðtþ1Þ
Ci
þg
þ Rg
;
(9)
where Cir is the number of POIs being assigned to region r
for user i, and Ci is the number of all POIs and all regions
for user i.
After updating region Rðtþ1Þ , we update Cðtþ1Þ by maximizing the posterior with respect to latent factors u and v .
We use a gradient ascent method to find Cðtþ1Þ that maximizes the posterior. Note that we already update R as
Rðtþ1Þ , and we here need to maximize the posterior with
respect to latent factor variables u and v . More specifically,
we maximize the following objective function:
LðU
U ; V j Rðtþ1Þ Þ ¼
M X
K
X
ððaU 1Þ ln uik uik =bU Þ
i¼1 k¼1
þ
N X
K
X
ððaV 1Þ ln vjk vjk =bV Þ
j¼1 k¼1
þ
where fij ¼
u>
i vj
M X
N
X
ðyij ln fij fij Þ þ const:
i¼1 j¼1
d0
½d0 þdði;jÞt .
(10)
LIU ET AL.: A GENERAL GEOGRAPHICAL PROBABILISTIC FACTOR MODEL FOR POINT OF INTEREST RECOMMENDATION
1173
Fig. 4. POI geographical distribution for the three different datasets.
Taking derivatives on L with respect to uik and vjk , we
have
t
N X
@L
aU 1
1
yij
d0
¼
þ
1 vjk
@uik
uik
bU j¼1 fij
d0 þ dði; jÞ
t
M X
@L
aV 1
1
yij
d0
¼
þ
1 uik
:
@vjk
vjk
bV
fij
d0 þ dði; jÞ
i¼1
(11)
We use stochastic gradient ascent to update uik and uik . Stochastic gradient ascent (descent) have been widely used for
many machine learning tasks [4]. The main process involves
randomly scanning all training instances and iteratively
updating parameters,
uik
uik þ @L
; vjk
@uik
vjk þ @L
;
@vjk
(12)
where is a learning rate.
Remark. The region R is updated in each E-step. The
latent factor model parameters are updated based on the
new regions. We summarize the parameter estimation procedure for Geo-PFM in Algorithm 1.
Algorithm 1. Geo-PFM Estimation
1: Initialize region partition Rð0Þ by k-means (k ¼ R)
2: for t
1 to Niteration do
3: Update region RðtÞ according to Equ. (7)
ðtÞ
4: Update region mean mðtÞ
r and covariance S r according to
Eq. (8)
ðtÞ
5: Update user region preference distribution h ir according
to Eq. (9)
ðtÞ ðtÞ
6: Update uik ; vjk by stochastic ascent
7: while not converge do
n
8:
nIter :¼ nþnIter1
//annealing learn rate
9:
for each random fi; jg pair do
10:
for k
1 to K do
uik þ nIter @u@L
11:
uik
mobility. One way to combine the predicted value and user
mobility is y^ij ¼ Eðyij jui ; vj Þ hir with j 2 r, the larger
the predicted value, the more likely the user will choose
this POI.
5
EXPERIMENTAL RESULTS
In this section we empirically evaluate the performance of
our proposed methods. All experiments were performed on
three real-world LBSN datasets, collected from Foursquare
(one of the most popular LBSN communities), Gowalla, and
Brightkite.
5.1 Datasets
Foursquare dataset. The Foursquare dataset is formulated as
follows [8], [9]: Foursquare users usually report their checkins at POIs via Twitter. When an LBSN user posts a Tweet
check-in at a POI, we consider it as evidence that the user
has physically checked into the POI. The dataset includes
POIs across the Unites States (except Hawaii and Alaska),
and the geographical distribution of all POIs is shown in
Fig. 4a. According to the Twitter reports from Foursquare
users, we finalized a dataset of 12;422 users for 46;194 POIs
with 738;445 check-in observations. The user POI check-in
count matrix has a sparsity of 99:87 percent; it is very sparse.
Each user checked into 59:44 POIs on average, only a very
small fraction of all the POIs. The number of check-ins for a
POI ranges from 1 to 786. This range is very wide as shown
in Fig. 5, in which the user check-in count of a randomly
chosen user is plotted.
Gowalla dataset. Besides the Foursquare dataset, we also
evaluate the proposed models on Gowalla [10]. In this dataset, we remove those POIs with less than 10 users, and
remove users with less than 30 user-POI pairs. We finalize a
dataset of 7;070 users for 30;755 POIs with 520;950 check-in
observations. The user POI check-in count matrix has a
ik
12:
vjk
vjk þ nIter @v@L
jk
13:
end for
14:
end for
15: end while
16: end for
4.3 Recommendation
After parameters C are learned, the Geo-PFM model predicts the check-in counts of a user for a given POI as
d0
t
Eðyij jui ; vj Þ ¼ u >
i v j ½d0 þdði;jÞ . We make recommendations
based on the predicted check-ins as well as the user
Fig. 5. An example of wide range user check-in counts for a randomly
chosen user (Foursquare).
1174
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TABLE 2
Data Description
Foursquare
Gowalla
Brightkite
# users
# POIs
# records
sparsity
avg POIs
12,422
7,070
2,192
46,194
30,755
9,865
738,445
520,950
72,543
99.87%
99.76%
99.66%
59.44
73.68
33.09
sparsity of 99:76 percent, with each user checked into 73:68
POIs on average. The number of check-ins for a POI ranges
from 1 to 286, and the geographical distribution of all
Gowalla POIs is shown in Fig. 4b.
Brightkite dataset. Finally, we evaluate the proposed models on Brightkite [10]. We finalize a dataset of 2;192 users
and 9;865 POIs with 72;543 check-in observations. The user
POI check-in count matrix has a sparsity of 99:66 percent,
with each user checked into 33:09 POIs on average. The
number of check-ins for a POI ranges from 1 up to more
than one thousand, and the geographical distribution of all
Brightkite POIs is shown in Fig. 4c. We summarize the data
statistics for all datasets in Table 2.
5.2 Evaluation Metrics
Since there is no explicit rating for validation, we evaluate
the models in terms of ranking. We present each user with
N POIs sorted by the predicted values and evaluate based
on which of these POIs were actually visited by the user.
Precision and recall. Given a top-N recommendation list
SN;rec sorted in descending order of the prediction values,
precision and recall are defined as
T
jSN;rec Svisited j
Precision@N ¼
TN
(13)
jSN;rec Svisited j
;
Recall@N ¼
jSvisited j
where Svisited are the POIs a user has visited in the test data.
The precision and recall for the entire recommender system
are computed by averaging all the precision and recall values of all the users, respectively.
F-measure. F-measure combines precision and recall, and
is the harmonic mean of precision and recall. Here we use
the Fb measure with b ¼ 0:5,
Fb ¼ ð1 þ b2 Þ Precision Recall
:
b Precision þ Recall
2
(14)
The Fb measure with b < 1 indicates more emphasis on precision than recall.
5.3 The Method for Comparison
We experimentally compare our proposed Poisson
Geo-PFM1 model with state-of-the-art latent factor models.
Specifically, we compare our proposed Poisson Geo-PFM
model with following algorithms:
Probabilistic Matrix Factorization [28]. PMF is a recommendation method widely used for different
1. We will refer to Poisson Geo-PFM as Geo-PFM in this Section
unless stated otherwise.
VOL. 27,
NO. 5,
MAY 2015
recommendation tasks, and the details of PFM are
summarized in Section 2.1.
Bayesian Non-negative Factorization (BNMF) [30]. This
is the base model which our earlier work [23]
adopted.
Poisson Factor Model [26]. Poisson factor model provides an alternative for count data recommendation
as Poisson is effective in modeling count data (more
details in Section 2.2).
Fused Poisson factor model (Fu-PoiFM). this method
fuses the geographical influence into factor models
by considering the multi-region of user check-in
behaviors and the inverse distance in an ad hoc way
[7]. Since Poison factor model also exploits the count
check-in characteristics, we fuse the geographical
influence into PoiFM and denote it as Fu-PoiFM.
Geo-BNMF. This is the model we used in our earlier
work [23].
In particular, we are interested in investigating the following questions:
How the proposed Geo-PFM improves the non-geographical baseline models (PMF, BNMF, PoiFM) as
well as the fused model (Fu-PoiFM).
How the Poisson based model Geo-PFM improves
its counterpart based on non-negative factorization,
Geo-BNMF.
We randomly divided the data into 80 percent for training and 20 percent for testing. We set U ¼ 0:005 and
V ¼ 0:005 for PMF. For Poisson factor based models used
in this experiment, we set aU ¼ 5; aV ¼ 20 and bU ¼
bV ¼ 0:2. We set 1=R for user region multinomial prior g.
d0
We set t ¼ 1 and d0 ¼ 0:2 for the distance model ½d þdði;jÞ
t .
0
For Geo-PFM and Fu-PoisonFM, we first cluster all the
POIs into jRj regions. This is the initialization of the
Geo-PFM model. We set the number of regions jRj ¼ 49,
which is the number of regions partitioned according to all
the states in USA (except Hawaii and Alaska). All the latent
factor models are implemented with stochastic gradient
ascent/descent optimization method with an annealing procedure to discount learning rate at iteration nIter with
n
by setting n ¼ 10.
nIter :¼ nþnIter1
5.4 Performance Comparison2
In this section, we present the performance comparison on
recommendation accuracy between our model and the baseline methods. We compare the results using both the
Foursquare and the Gowalla dataset by setting latent
dimensions to K ¼ 10 and K ¼ 20.
Foursquare dataset. Fig. 6 shows the precision and
recall@N (N ¼ 1; 5; 10) all the methods achieve on the Foursquare dataset, and Table 3 shows the Fb measure (b ¼ 0:5).
From the results, it is clear that the proposed Geo-PMF consistently outperforms all the baseline methods, including
the non-geographical baseline models (PMF, BNMF, PoiFM)
as well as the fused model (Fu-PoiFM). Specifically,
2. In the experiments of this paper, we rank the top-N recommendation globally, which is different from the regional way we used in [23].
Also we further tune some parameters. Therefore, the absolute experimental values in this paper may somewhat differ from those in [23].
LIU ET AL.: A GENERAL GEOGRAPHICAL PROBABILISTIC FACTOR MODEL FOR POINT OF INTEREST RECOMMENDATION
Fig. 6. Precision and Recall with two different latent dimensions K (Foursquare dataset). Note that we focus on two comparisons: (1) How the
proposed Geo-PFM improves the non-geographical baseline models
(PMF, BNMF, PoiFM) as well as the fused model (Fu-PoiFM); (2) How
the Poisson based model Geo-PFM improves its non-negative factorization based counterpart Geo-BNMF.
TABLE 3
Fb Measure (b ¼ 0:5) with Two Different Latent Dimensions K
(Foursquare Dataset)
K @N
PMF BNMF PoiFM Fu-PoiFM Geo-BNMF Geo-PFM
@1 0.0083 0.0087 0.0091
10 @5 0.0061 0.0100 0.0145
@10 0.0047 0.0090 0.0159
0.0150
0.0236
0.0242
0.0130
0.0263
0.0339
0.0220
0.0328
0.0339
@1 0.0087 0.0088 0.0095
20 @5 0.0123 0.0131 0.0151
@10 0.0092 0.0117 0.0162
0.0157
0.0241
0.0251
0.0141
0.0274
0.0296
0.0224
0.0346
0.0353
nonnegative based Poison factor model and BNMF outperform PMF. Furthermore, PoiFM outperforms its counterpart
BNMF by making Poisson assumptions. The fused method,
Fu-PoiFM, improves PoiFM due to the fusion of geographical influence and multi-center user activity pattern into the
latent factor model. Our proposed Geo-PFM further
improves Fu-PoiFM significantly. From Table 3, we can
observe an average of 0:0089 improvement in terms of Fb
measure for Geo-PFM over Fu-PoiFM.
Meanwhile, from Fig. 6 we can see that the Poisson-based
model Geo-PFM improves its counterpart based on nonnegative factorization, Geo-BNMF, with an average of
0:0069 improvement in terms of Fb measure. This improvement can be ascribed to the following reasons. First, the
Poisson-based latent factor is more appropriate for modeling count data. As shown, the improved performance of
PoiFM over BNMF from Fig. 6, PoiFM can improve BNMF
with an average of 0:0032 improvement in terms of Fb .
Second, the Poisson Geo-PFM provides a more rigorous
probabilistic generative process for the model, while the
non-negative matrix factorization based Geo-PFM applied
an approximation solution. As shown in the model
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Fig. 7. Precision and Recall with two different latent dimensions K
(Gowalla dataset).
TABLE 4
Fb Measure (b ¼ 0:5) with Two Different Latent Dimensions K
(Gowalla Dataset)
K @N
PMF BNMF PoiFM Fu-PoiFM Geo-BNMF Geo-PFM
@1 0.0091 0.0123 0.0135
10 @5 0.0272 0.0272 0.0298
@10 0.0207 0.0221 0.0322
0.0306
0.0629
0.0682
0.0338
0.0483
0.0551
0.0442
0.0759
0.0778
@1 0.0128 0.0119 0.0135
20 @5 0.0273 0.0290 0.0298
@10 0.0201 0.0295 0.0323
0.0310
0.0632
0.0681
0.0335
0.0491
0.0520
0.0442
0.0761
0.0779
estimation in Section 4.2, we need a rigorous probability for
model inference. While the Poisson based model provides
an exact probability representation, the Geo-BNMF applies
a rectified normal distribution.
Gowalla dataset. Fig. 7 shows the precision and recall@N
(N ¼ 1; 5; 10) of all the methods evaluated on the Gowalla
dataset, and the corresponding Fb measure values are
shown in Table 4. We can clearly observe that the proposed
Geo-PFM performs consistently better over all the baseline
methods. From Table 4, we can observe an average of
0:0121 improvement in terms of Fb measure for Geo-PFM
over Fu-PoiFM. We further observe that the Poisson-based
Geo-PFM improves Geo-BNMF by an average of 0:0207
increase in terms of Fb measure.
Brightkite dataset. Fig. 8 shows the precision and recall@N
(N ¼ 1; 5; 10) of all the methods evaluated on the Gowalla
dataset, and the corresponding Fb measure values are
shown in Table 4. We can still observe consistent improvements of the proposed Geo-PFM over all the baseline methods. From Table 5, we can observe an average of 0:0246
improvement in terms of Fb measure for Geo-PFM over
Fu-PoiFM. Again, we further observe that Poisson-based
Geo-PFM improves Geo-BNMF with an average of 0:0129
increase in terms of Fb measure.
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VOL. 27,
NO. 5,
MAY 2015
Fig. 9. Voronoi visualization of POI segmentation in California area
(Foursquare): (b) latent regions learned by Geo-PFM, (a) initiation by
K-means, and (c) true user collaborative activity clusters. Deeper color
(red) indicates more check-ins for a POI, as contrary to light color
(green). Best view in color.
Fig. 8. Precision and Recall with two different latent dimensions K
(Brightkite dataset).
TABLE 5
Fb Measure (b ¼ 0:5) with Two Different Latent Dimensions K
(Brightkite Dataset)
K @N
PMF BNMF PoiFM Fu-PoiFM Geo-BNMF Geo-PFM
@1 0.0092 0.0140 0.0188
10 @5 0.0088 0.0186 0.0241
@10 0.0067 0.0221 0.0238
0.0439
0.0383
0.0337
0.0513
0.0553
0.0558
0.0699
0.0612
0.0632
@1 0.0110 0.0226 0.0252
20 @5 0.0112 0.0276 0.0387
@10 0.0094 0.0269 0.0336
0.0453
0.0501
0.0438
0.0508
0.0553
0.0564
0.0729
0.0682
0.0670
Comparisons across different datasets. First, we observed
consistent improvements of the proposed Geo-PFM over all
the baseline methods, though the three dataset differ in terms
of user-POI observation sparsity, response skewness, and
POI geographical distributions (see Fig. 4). Second, the Poisson-based Geo-PFM improves its counterpart based on nonnegative factorization, Geo-BNMF. Third, user-POI observation sparsity, response skewness and POI geographical distributions could affect the algorithm performances. For
example, the results on the Gowalla dataset and the Brightkite dataset are better than those on the Foursquare dataset.
The Gowalla dataset is much denser than the Foursquare
dataset. Note that Gowalla dataset has a sparsity of 99:76 percent, and an average of 59:44 user-POI observations; while
the Foursquare dataset has a sparsity of 99:87 percent, an
average of 73:68 user-POI observations. Although Brightkite
dataset has fewer user-POI observations, on average, than
Foursquare dataset, its sparsity is the lowest among the three
datasets. Further, the Gowalla dataset is less skewed than
the Foursquare dataset. These two factors could allow the
latent factor models, both PMF and PoiFM, to achieve better
performances. Also, the Gowalla dataset is more geographically centralized than the Foursquare dataset. As a result, the
performances of Geo-PFM would be more obvious
compared to Fu-PoiFM when applied to more geographically distributed circumstances.
Latent region analysis. In addition to improving recommendation performance, our proposed model also provides
a unique perspective on POI marketing segmentation, in the
form of the learned regions. We take a representative area,
California, as an example to analyze the regions learned by
the Geo-PFM model. Fig. 9 visualizes the latent regions
(Fig. 9b) learned from our model in versus its initialization
by K-means (Fig. 9a) in California. Though we have no
ground truth about an optimal POI region segmentation,
we can infer the user activity regions in California through
the collective check-in behaviors of users who have visited
California and view the region clusters formulated by collective check-ins as ground truth (see Fig. 9c). Through analyzing the collaborative check-in frequency by those users,
as shown in Fig. 9c, we can see two clear clusters in northern
California among other scattered POIs, one cluster in the
Los Angeles area, one in San Diego, and some scattered
POIs between southern and northern California. K-means
only depends on POI distances to cluster POIs for region
segmentation. As shown in Fig. 9a, K-means segments
northern California into four different regions, and segments Los Angeles into two regions. However, by considering the user check-in behaviors and geographical factors,
our model identified a more meaningful region partition as
shown in Fig. 9b, which is more coherent to real user activity as shown in Fig. 9c. Geo-PFM initiated by K-means
leads to better POI segmentation. We can see that Geo-PFM
models not only improve recommendation performance,
but also provide an interesting perspective on POI marketing segmentation in the form of the learned regions.
Summary. Geographical influence and user mobility are
two of the most important characteristics for LBSNs, and
play an important role in POI recommendation. The fused
method (Fu-PoiFM) which exploits an ad hoc two-step process to fuse the geographical influence and multi-center
user activity pattern into user preferences can improve pure
latent factor model (PoiFM). However, an integrated analysis of multiple factors for POI recommendations lead to further improvements. The proposed Geo-PFM model not
only considers the geographical information of POIs and
user mobility patterns for recommendation, but also
updates the latent regions by considering these sources of
information. The learned regions reflect the collaborative
LIU ET AL.: A GENERAL GEOGRAPHICAL PROBABILISTIC FACTOR MODEL FOR POINT OF INTEREST RECOMMENDATION
user activity pattern. As a result, we can observe obvious
improvements over all the baseline algorithms. Also, as
shown in the performance of Poisson factor model compared to its Gaussian counterpart PFM, we observe
improvements by Poison factor model, as Poisson distribution is more suitable for modeling count data. Further evidence of this is the fact that the Poisson based model
Geo-PFM improves its non-negative factorization based
counterpart Geo-BNMF in all the evaluation datasets,
though Geo-BNMF imposes a non-negativity constrain.
6
RELATED WORK
Recommender systems can be developed based on explicit
user feedback. In other words, users rate items and the
user-item preference relationship can be modeled on the
basis of the user ratings. Latent factor models, such as as
matrix factorization [19], probabilistic matrix factorization
[28], its non-parametric version [27], and other other variants [1], [3], [17], [18], [22], [36], have become popular and
widely used in recommendation. Most of the latent factors
along this line of work assume that the response follows a
Gaussian distribution over the product of user and item
latent factors. The Gaussian-based latent factor models can
achieve good prediction performance when explicit ratings
are available. In contrast, recommender systems can also be
developed based on implicit user feedback [16], such as the
search and click behaviors on a web site [26], advertisement
targeting [6], and the check-in behaviors in LBSNs, as we
discussed in this paper. In this case, the recommender system has to infer user preferences from implicit user feedback. Here, latent factor models which are suitable for
implicit user feedback are preferred. One option is to set
non-negative constraints on latent factors to force the
response variable into a wider range than the rating-based
response. As a result, methods based on non-negative
matrix factorization are widely used [13], [21], [25], [37].
However, the Poisson distribution is suitable for modeling
count data. As a result, Poisson factor models are widely
used for count based feedback recommendation settings [5],
[6], [12], [26].
Some previous studies on POI recommendation, or more
precisely location recommendation, mainly relied on user
trajectory data to infer user preferences. For example, previous works [11], [38], [39], [40], [41] applied collaborative filtering (CF) methods to recommend locations and taxi pickup locations based on user trajectory data. However, POI
recommendation provide exact POIs a user would be interested rather than a “location”. Due to the development and
popularity of location-based social networks, more recent
works, such as [33], [34], began to explore user preferences,
social influence, and geographical influence for POI recommendations. However, these used a simple CF algorithm to
fuse this information, and thus lack a comprehensive way
to model how all this information collectively influence user
POI check-in decision. The work in [24] tried to explore side
information to improve POI recommendations, but it does
not explore user mobility information and does not take the
skewed data characteristics of implicit user check-in counts
into the consideration. Kurashima et.al [20] extended the
latent Dirichlet allocation (LDA) model to include
1177
geographical influence to profile user location preference,
but it did not consider user mobility and the user activity
areas modeled in this paper are constrained only to areas
that a user has traveled to.
More recently, Cheng et al. [7] considered the geographical influence, the multi-center of user check-in patterns, the
skewed user check-in frequency and social networks for
POI recommendation. However, this work applied an ad hoc
two-step method to fuse the geographical influence into
user preferences, and did not really consider the user mobility and lacked an integrated consideration of factors that can
influence POI recommendation. Moreover, the greedy clustering method applied to derive the personalized multi-centers could easily lead to overfitting problems in that it
focuses on the regions a user has visited. Instead, our work
is an integrated analysis of geographical influences, user
mobility, and skewed data for POI recommendation.
Hu and Ester [15] proposed a spatial topic model by considering the spatial and textual aspects of posts published by
mobile users, and predict future user locations as POI recommendation. This is the work most closely related to ours
in terms of the way to account for geographical influence
and user mobility. However, their work is more similar to a
location prediction problem than a POI recommendation
task. Moreover, the Poisson model used in this paper could
be equivalent conditioned on the per-user sums and where
the item weights are constrained to sum to one [12], [42],
[43]. However, our proposed Geo-PFM is more flexible and
can be extended to different latent factor settings.
In addition, our work has a connection with recent works
on mobility modeling [10], [14]. However, their tasks were
different. Work [14] used a similar multinomial assumption
over different regions to model geographical topics in Twitter stream, and the work in [10] investigated human mobility for social network analysis. Also, people have used
Gaussian distribution to model region over locations [14],
[31], [35].
As described above, while there are some studies on POI
recommendation, they lacks an integrated analysis of the
joint effects of multiple factors that influence the decision
process of a user choosing a POI. These factors include user
interest preferences, geographical influences, user mobility
pattern, and the skewed implicit user check-in count data.
The proposed method strategically takes all these factors
into consideration and presents a flexible probabilistic generative model for POI recommendations.
7
CONCLUSION AND DISCUSSION
In this paper, we presented an integrated analysis of the
joint effect of multiple factors which influence the decision
process of a user choosing a POI and proposed a general
framework to learn geographical preferences for POI recommendation in LBSNs. The proposed geographical probabilistic factor analysis framework strategically takes all these
factors, which influence the user check-in decision process,
into consideration. There are several advantages of the proposed recommendation method. First, the model captures
the geographical influence on a user’s check-in behavior by
taking into consideration the geographical factors in LBSNs,
such as the Tobler’s first law of geography. Second, the
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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING,
methods effectively modeled the user mobility patterns,
which are important for location-based services. Third, the
proposed approach extended the latent factors from explicit
rating recommendation to implicit feedback recommendation settings by considering the skewed count data characteristic of LBSN check-in behaviors. Last but not least, the
proposed model is flexible and could be extended to
incorporate different latent factor models, which are suitable for both explicit and implicit feedback recommendation settings. Finally, extensive experimental results on realworld LBSNs data validated the performance of the
proposed method.
Limitations and discussion. Geographical influence and
user mobility are among the most important characteristics
in LBSNs and could greatly affect POI recommendation.
The proposed Geo-PFM model captures these two factors
by introducing latent regions, which represent the collective
user activity areas. This method coarsely captures the geographical influence and user mobility. However, the geographical influence and user mobility can be subtle [10],
[29]. A possible future direction is to combine both the macroscopic and microscopic effects of geographical influence
and user mobility.
ACKNOWLEDGMENTS
This is a extended and revised version of [23], which
appears in the Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining (KDD
2013). This research was partially supported by US National
Science Foundation (NSF) via grant numbers CCF-1018151
and IIS-1256016. Also, it was supported in part by Natural
Science Foundation of China (71028002). The authors would
like to thank the anonymous reviewers for their valuable
comments and suggestions to improve the quality of the
paper.
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Bin Liu is currently working toward the PhD
degree in the Department of Management Science and Information Systems, Rutgers Business
School, Rutgers University. His general area of
research is data mining and business analytics,
with a focus on recommender systems, mining
rich user-generated content, and location-based
services.
Hui Xiong (SM’07) received the BE degree from
the University of Science and Technology of
China (USTC), China, the MS degree from
the National University of Singapore (NUS),
Singapore, and the PhD degree from the University of Minnesota (UMN). He is currently a professor and vice chair of the Management Science
and Information Systems Department, and the
director of Rutgers Center for Information Assurance at the Rutgers, the State University of New
Jersey, where he received a two-year early promotion/tenure in 2009, the Rutgers University Board of Trustees
Research Fellowship for Scholarly Excellence in 2009, and the ICDM2011 Best Research Paper Award in 2011. His general area of research
is data and knowledge engineering, with a focus on developing effective
and efficient data analysis techniques for emerging data intensive applications. He has published prolifically in refereed journals and conference
proceedings (three books, more than 40 journal papers, and more than
60 conference papers). He is a coeditor-in-chief of Encyclopedia of GIS,
an associate editor of IEEE Transactions on Data and Knowledge Engineering (TKDE), and the Knowledge and Information Systems (KAIS)
journal. He has served regularly on the organization and program committees of numerous conferences, including as a program co-chair of the
Industrial and Government Track for the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining and a program co-chair for the 2013 IEEE International Conference on Data
Mining (ICDM-2013). He is a senior member of the ACM and IEEE.
1179
Spiros Papadimitriou is an assistant professor
in the Department of Management Science &
Information Systems at Rutgers Business
School. Previously, he was a research scientist
at Google, and a research staff member at IBM
Research. His main interests are large scale data
analysis, time series, graphs, and clustering. He
has published more than 40 papers on these
topics and has three invited journal publications
in best paper issues, several book chapters and
he has filed multiple patents. He has also given a
number of invited talks, keynotes, and tutorials. He received the Siebel
Scholarship in 2005 and the Best Paper Award in SDM 2008.
Yanjie Fu received the BE degree from the University of Science and Technology of China,
China, 2008, the MS degree from the Chinese
Academy of Sciences, China, 2011. He is currently working toward the PhD degree in the Management Science and Information Systems
Department at Rutgers University. His research
interests include data mining, business analytics,
geoeconomics, and customer targeting.
Zijun Yao received the BE degree in electrical
engineering from the Guangdong University of
Technology in 2009, the MS degree in computer engineering from Northeastern University
in 2011. He is currently working toward the
PhD degree in information technology at
Rutgers, The State University of New Jersey.
His research interests include data mining and
business analytics.
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